STAT 400 Chap.6 Joint Distributions, Covariance and Correlation
Joint Distributions, Covariance and Correlation
Two variables in one table
A joint probability mass function assigns a probability to every pair of values of two discrete random variables, and for STAT 400 it is usually displayed as a table. The row and column totals are the marginal distributions, which describe each variable on its own. Dividing a row by its total gives a conditional distribution, the distribution of the second variable once the first is known.
Reading a table in these three ways is the core skill of the chapter.
Independence cell by cell
Two random variables are independent when every joint probability equals the product of the matching marginals. The check is demanding: a single cell that fails the product rule proves dependence, while proving independence needs every cell.
A quick visual clue is that independent variables have proportional rows, because each row is the same conditional distribution scaled by its marginal probability.
Covariance and correlation
Covariance measures how two variables move together and is easiest to compute as E[XY] − E[X]E[Y]. Its sign shows the direction of a linear trend, but its size depends on the units.
Correlation divides by both standard deviations to give a unit-free number between −1 and 1, with the extremes reached only by exact straight-line relationships.
Independent variables have zero covariance, but zero covariance does not imply independence: a variable and its square can have zero covariance while being completely dependent, because covariance only detects linear association.
Linear combinations
The variance of aX + bY is a²Var(X) + b²Var(Y) + 2ab·Cov(X, Y). The covariance term is where most mistakes happen.
For independent variables it vanishes, and then the variance of a difference equals the variance of a sum: variances add in both cases.
The same formula extends to many variables and underlies the next chapter, where the variance of a sample mean is σ²/n.
Reading a correlation value
A correlation near 0.4, as in a moderately related pair of scores, says that above-average values of one variable tend to come with above-average values of the other, but with a lot of scatter.
It does not say that one causes the other, and it says nothing about curved relationships.
Because correlation has no units, rescaling either variable, for example converting minutes to hours, leaves it unchanged, while the covariance changes by the same factor as the rescaling.
How it is assessed
The course objectives ask you to compute probabilities from a discrete bivariate table, to explain the terms joint and marginal, and to calculate and interpret covariance and correlation.
Expect a table-reading question that ends with an interpretation sentence.
What this chapter covers
- 01
Joint probability tables
- 02
Marginal and conditional distributions
- 03
Independence of random variables
- 04
Covariance and correlation
- 05
Variance of linear combinations
Worked example · free
Covariance from a small joint table
- 2Marginals: P(X = 1) = 0.50 and P(Y = 1) = 0.55.
- 2E[XY] = P(1,1) = 0.40, so Cov(X, Y) = 0.40 − 0.50(0.55) = 0.125.
- 2Independence fails: P(1,1) = 0.40 but P(X = 1)P(Y = 1) = 0.275.
Key terms
- Joint PMF
- The function giving P(X = x, Y = y) for every pair of values of two discrete random variables.
- Marginal Distribution
- The distribution of one variable alone, obtained by summing the joint pmf over the other variable.
- Covariance
- The expected product of deviations from the means, E[XY] − E[X]E[Y], measuring linear co-movement in the variables' units.
- Correlation Coefficient
- Covariance divided by the product of the standard deviations, a unit-free measure of linear association between −1 and 1.
Joint Distributions, Covariance and Correlation FAQ
Does zero correlation mean two variables are independent?
No. Zero correlation rules out a linear trend only. A variable and its square, for a distribution symmetric about zero, have zero correlation even though one determines the other.
Why do variances add for X − Y?
The coefficient of Y is −1, and it enters the variance formula squared, giving +1. Only the covariance term changes sign, and it is zero for independent variables.
How do I get a conditional distribution from a joint table?
Take the row or column for the value you are conditioning on and divide each entry by that row or column total, so the conditional probabilities sum to one.
Exam move
Build two joint tables yourself, one independent and one dependent, and compute marginals, covariance and correlation for each. Then write one sentence interpreting the sign and size of each correlation in context.
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